Grand-Canonical Typicality

Abstract

Abstract We study how the grand-canonical density matrix arises in macroscopic quantum systems. “Canonical typicality” is the known statement that for a typical wave function $$\Psi $$ Ψ from a micro-canonical energy shell of a quantum system S weakly coupled to a large but finite quantum system B , the reduced density matrix $$\hat{\rho }^S_\Psi ={{\,\textrm{tr}\,}}^B |\Psi \rangle \langle \Psi |$$ ρ ^ Ψ S = tr B | Ψ ⟩ ⟨ Ψ | is approximately equal to the canonical density matrix $$\hat{\rho }_\textrm{can}=Z^{-1}_\textrm{can}\exp (-\beta \hat{H}^S)$$ ρ ^ can = Z can - 1 exp ( - β H ^ S ) . Here, we discuss the analogous statement and related questions for the grand-canonical density matrix $$\hat{\rho }_\textrm{gc}=Z^{-1}_\textrm{gc}\exp (-\beta (\hat{H}^S-\mu _1 \hat{N}_{1}^S-\ldots -\mu _r\hat{N}_{r}^S))$$ ρ ^ gc = Z gc - 1 exp ( - β ( H ^ S - μ 1 N ^ 1 S - … - μ r N ^ r S ) ) with $$\hat{N}_{i}^S$$ N ^ i S the number operator for molecules of type i in the system S . This includes (i) the case of chemical reactions (which requires some novel considerations) and (ii) that of systems S defined by a spatial region which particles may enter or leave. It includes statements about how $$\hat{\rh